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The long exact homology sequence is natural
Statement
A morphism of short exact sequences of complexes induces a morphism between the associated long exact homology sequences.
Facts & Assumptions
Given: A morphism of short exact sequences of complexes.
Every chain map induces a well-defined map on homology (A chain map induces a well-defined map on homology).
The connecting square commutes under a morphism of short exact sequences (Naturality of the homology connecting morphism).
Proof
Each ordinary square in the two long exact sequences is induced by one of the three chain maps in the given ladder, so it commutes by [L1].
The only nonformal squares are the connecting ones, and they commute by [L2]. Therefore every square in the long exact ladder commutes.
Depends on
Used by
- Two-out-of-three for quasi-isomorphisms in a short exact sequence diagram Corollary
- FALSE: naturality of the long exact sequence follows without checking the connecting square False statement
- Coordinate-ball classes identify local homology stalks Lemma
- Refinement-limit Cech cohomology computes sheaf cohomology on compact Hausdorff spaces Lemma
- The Kunneth Tor map Lemma
- An exact functor carries the long exact homology sequence to the corresponding long exact sequence Proposition
- Homology of complexes satisfies the delta-functor naturality and exactness laws Proposition
- Left derived functors form a homological delta functor Theorem
- Naturality of singular Mayer–Vietoris Theorem
- Naturality of the pair long exact sequence Theorem
- The natural PID Kunneth sequence is exact Theorem
- Top homology of a connected manifold Theorem
Dependency tree · two levels
14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
- The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)